What is the time complexity of Best Time to Buy and Sell Stock with Cooldown?

The optimal solution has a time complexity of O(n).

What data structures are used to solve Best Time to Buy and Sell Stock with Cooldown?

This problem uses Array, Dynamic Programming. Understanding these concepts is essential for solving this Medium problem efficiently.

What are the key insights for solving Best Time to Buy and Sell Stock with Cooldown?

Insight 1: Dynamic Programming is the appropriate technique to solve this optimization problem, allowing us to build up the solution from subproblems. Insight 2: We need to track three states for each day: the maximum profit if we 'buy' on that day, the maximum profit if we 'sell' on that day, and the maximum profit if we are in a 'cooldown' state.

Which companies ask LeetCode 309. Best Time to Buy and Sell Stock with Cooldown in interviews?

This problem is frequently asked at Geico, Google, Visa.

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Best Time to Buy and Sell Stock with Cooldown - LeetCode 309 Solution

Best Time to Buy and Sell Stock with Cooldown - Complete Solution Guide

Best Time to Buy and Sell Stock with Cooldown is LeetCode problem 309, a Medium level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

You are given an array prices where prices[i] is the price of a given stock on the i th day. Find the maximum profit you can achieve. You may complete as many transactions as you like (i.e., buy one and sell one share of the stock multiple times) with the following restrictions: After you sell your stock, you cannot buy stock on the next day (i.e., cooldown one day). Note: You may not engage in multiple transactions simultaneously (i.e., you must sell the stock before you buy again). Example 1:

Detailed Explanation

The problem asks us to find the maximum profit we can make by buying and selling a stock multiple times, given an array of stock prices for each day. However, there's a cooldown period of one day after each sale, meaning you cannot buy stock on the day immediately following a sale. You also cannot engage in multiple transactions simultaneously, implying you must sell before you buy again. The input is an array `prices` representing the stock price on each day, and the output is the maximum profit achievable.

Solution Approach

The solution uses dynamic programming with state compression to achieve O(1) space complexity. Instead of using arrays to store the maximum profit for buy, sell, and cooldown states at each day, it uses three variables `buy_prev`, `sell_prev`, and `sell_prev_prev` to store the maximum profit for these states at the previous and two previous days. This approach works by iteratively updating these variables as we traverse the `prices` array. The `buy_curr` and `sell_curr` are computed based on previous values ensuring the cooldown constraint is respected.

Step-by-Step Algorithm

Step 1: Initialize `buy_prev` to negative infinity (Integer.MIN_VALUE in Java/C++) because initially, we cannot have any profit if we buy the stock on the first day. Initialize `sell_prev` and `sell_prev_prev` to 0 because if we don't buy or sell anything, the profit is 0.
Step 2: Iterate through the `prices` array, where each `price` represents the stock price on a given day.
Step 3: Calculate `buy_curr`. The maximum profit when buying on the current day is the maximum of: a) the profit from buying on the previous day (`buy_prev`), and b) the profit from selling two days ago minus the current price (`sell_prev_prev - price`). This ensures the cooldown period is enforced, as we're buying after a cooldown.
Step 4: Calculate `sell_curr`. The maximum profit when selling on the current day is the maximum of: a) the profit from selling on the previous day (`sell_prev`), and b) the profit from buying on the previous day plus the current price (`buy_prev + price`).
Step 5: Update the variables for the next iteration. `sell_prev_prev` becomes `sell_prev`, `sell_prev` becomes `sell_curr`, and `buy_prev` becomes `buy_curr`.
Step 6: After iterating through all prices, the final `sell_prev` will contain the maximum profit we can achieve. Return `sell_prev`.

Key Insights

Insight 1: Dynamic Programming is the appropriate technique to solve this optimization problem, allowing us to build up the solution from subproblems.
Insight 2: We need to track three states for each day: the maximum profit if we 'buy' on that day, the maximum profit if we 'sell' on that day, and the maximum profit if we are in a 'cooldown' state.
Insight 3: Because of the cooldown, the 'buy' state on a particular day depends on the 'sell' state from *two* days prior, not just the previous day.

Complexity Analysis

Time Complexity: O(n)

Space Complexity: O(1)

Topics

This problem involves: Array, Dynamic Programming.

Companies

Asked at: Geico, Google, Visa.

Best Time to Buy and Sell Stock with Cooldown | LeetCode 309

Medium

Array Dynamic Programming

Time: O(n)Space: O(1)

Solution Code

Available in:

class Solution {
    public int maxProfit(int[] prices) {
        int buy_prev = Integer.MIN_VALUE;
        int sell_prev = 0;
        int sell_prev_prev = 0;

        for (int price : prices) {
            int buy_curr = Math.max(buy_prev, sell_prev_prev - price);
            int sell_curr = Math.max(sell_prev, buy_prev + price);

            sell_prev_prev = sell_prev;
            sell_prev = sell_curr;
            buy_prev = buy_curr;
        }

        return sell_prev;
    }
}

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Problem Info

Description

Problem 309: Best Time to Buy and Sell Stock with Cooldown - Detailed explanation and solution approaches.

Topics

Array Dynamic Programming

Companies

GeicoGoogleVisa

Stats

Difficulty:Medium

Languages:4

Companies:3

Problem #:309

Problem Explanation

💡

Think Before You Code

🤔 Ask Yourself

→What is the input? What output do I need?
→Can I solve a smaller version of this problem first?
→What patterns have I seen in similar problems?

✏️ Try This First

Before looking at the solution, try to solve it on paper with a simple example!

🚀 Try on LeetCode

Solution Approach

📋

Step-by-Step Solution

6 steps

1
Step 1: Initialize `buy_prev` to negative infinity (Integer.MIN_VALUE in Java/C++) because initially, we cannot have any profit if we buy the stock on the first day. Initialize `sell_prev` and `sell_prev_prev` to 0 because if we don't buy or sell anything, the profit is 0.
2
Step 2: Iterate through the `prices` array, where each `price` represents the stock price on a given day.
3
Step 3: Calculate `buy_curr`. The maximum profit when buying on the current day is the maximum of: a) the profit from buying on the previous day (`buy_prev`), and b) the profit from selling two days ago minus the current price (`sell_prev_prev - price`). This ensures the cooldown period is enforced, as we're buying after a cooldown.
4
Step 4: Calculate `sell_curr`. The maximum profit when selling on the current day is the maximum of: a) the profit from selling on the previous day (`sell_prev`), and b) the profit from buying on the previous day plus the current price (`buy_prev + price`).
5
Step 5: Update the variables for the next iteration. `sell_prev_prev` becomes `sell_prev`, `sell_prev` becomes `sell_curr`, and `buy_prev` becomes `buy_curr`.
6
Step 6: After iterating through all prices, the final `sell_prev` will contain the maximum profit we can achieve. Return `sell_prev`.

💡

Pro Tip for Students

Try tracing through each step with a small example on paper before coding. Understanding "why" each step happens is more important than memorizing the code!

⚡

Complexity Analysis

⏱️

Time Complexity

O(n)

👍 Linear - Good! Visits each element once.

💾

Space Complexity

O(1)

💪 Constant space - Uses same memory regardless of input!

📝

Why?

The algorithm's time complexity is determined by the single loop that iterates through the input array. The space complexity is constant because the amount of memory used does not grow with the size of the input array. This is achieved by state compression.

📖New to Big O? Click to learn more

Big O Notation - Quick Guide

O(1)Best - Instant

O(log n)Excellent - Binary Search

O(n)Good - One loop

O(n log n)OK - Sorting

O(n²)Slow - Nested loops

O(2ⁿ)Very Slow - Recursion

Alternative Approaches

Dynamic Programming with Array

Using three arrays `buy[i]`, `sell[i]`, and `cooldown[i]` to store the maximum profit at day `i` for each of the three states. This allows for a more intuitive understanding of the state transitions.

Time: O(n), where n is the number of days.Space: O(n), because we are using three arrays of size n.

Trade-offs: This approach is easier to understand but uses O(n) space compared to the O(1) space of the main solution. This is useful when memory is not a big concern and readability is more important.

✨

Key Insights

1Insight 1: Dynamic Programming is the appropriate technique to solve this optimization problem, allowing us to build up the solution from subproblems.
2Insight 2: We need to track three states for each day: the maximum profit if we 'buy' on that day, the maximum profit if we 'sell' on that day, and the maximum profit if we are in a 'cooldown' state.
3Insight 3: Because of the cooldown, the 'buy' state on a particular day depends on the 'sell' state from *two* days prior, not just the previous day.

📚

Key Terms Explained

For Students

Array

A collection of items stored at contiguous memory locations. Like a row of boxes.

Dynamic Programming

Breaking complex problems into smaller subproblems and storing results.

Edge Cases

Edge case 1: `prices` is empty or contains only one element. The loop won't execute, and the initial value of `sell_prev` (0) will be returned, which is the correct answer.
Edge case 2: `prices` is sorted in descending order. In this case, it's never profitable to buy the stock, and the algorithm will correctly return 0 as the maximum profit.
Edge case 3: Very large price fluctuations. The `Integer.MIN_VALUE` or `-float('inf')` are chosen as the initial values of `buy_prev` so that the initial calculation would result in `sell_prev_prev - price`. The code effectively handles negative values.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Not considering the cooldown period. Forgetting that after selling on day i, you cannot buy on day i+1.

✗Common mistake 2: Trying to solve it greedily. A greedy approach might not yield the optimal solution due to the cooldown constraint. For example, always buying when prices are low and selling when they are high will fail in some scenarios.

✗Common mistake 3: Incorrectly initializing `buy_prev`. It needs to be initialized to a very low value (`-Infinity`) to reflect the fact that you cannot have profit on the first day if you buy.

✗Common mistake 4: Confusing the order of updates. `sell_prev_prev`, `sell_prev`, and `buy_prev` needs to be updated in correct order for the calculation to work.

✅

Learning Tip

Every mistake is a learning opportunity! When you make an error, understand why it happened before moving on.

🌍

Where is This Used in Real Life?

💼Real-world scenario 1: Algorithmic trading. The same principles can be applied to develop automated trading strategies for stocks, commodities, or currencies, taking into account transaction costs and trading restrictions.

🏦Real-world scenario 2: Resource allocation with constraints. The algorithm can be adapted to solve resource allocation problems with limitations on when resources can be used, such as scheduling tasks with dependencies and cooldown periods.

💡Understanding real-world applications helps you see the "why" behind algorithms. It's not just about passing interviews—these concepts power the apps and websites you use every day!

Author's Notes

💡 Key Insight

When dealing with arrays, always consider if sorting would simplify the problem. Many array problems have O(n) solutions using hash maps or two-pointer technique.

🎯 Interview Tip

Medium problems are the bread and butter of technical interviews. Practice explaining your approach before coding.

⚠️ Watch Out

Off-by-one errors are the most common mistake. Double-check your loop bounds and array indices.

Written by Saswata Mondal

Software Engineer • Algorithms Enthusiast • Interview Coach

👨‍💻

About the Author

👨‍💻

Saswata MondalAuthor400+ LC Problems

B.Tech CSE, IEM Kolkata | LeetCode Rating: 1672 (Top 15%)

Certified: IBM Java Developer, NPTEL Programming in Java

Last updated: October 8, 2024LeetCode Profile GitHub

This solution has been crafted based on my experience solving 400+ LeetCode problems. Each explanation is designed to help you understand the underlying concepts, not just memorize the code.

Community Tips & Insights

In a real interview, always clarify constraints first. Ask about input size and value ranges.

@TechLeadJul 14

Don't forget to handle the edge case when the input is empty. Many candidates miss this in interviews.

@DevExpert92Jul 7

The key insight here is recognizing the pattern. Once you see it, the solution becomes straightforward.

@AlgoMasterJun 30

❓

Frequently Asked Questions

When should I use in-place modification vs creating a new array?

Use in-place modification when space complexity matters (O(1) space requirement) and the original array can be modified. Create a new array when you need to preserve the original data or when the problem involves significant restructuring that would complicate in-place logic.

Should I use top-down (memoization) or bottom-up (tabulation)?

Start with top-down (recursion + memoization) as it directly maps to the recursive thinking and handles only needed subproblems. Convert to bottom-up for better space efficiency and to avoid stack overflow. Bottom-up is often faster due to no recursion overhead.

How much time should I spend on a medium problem in interviews?

In a 45-minute interview, aim for: 5-10 minutes understanding and clarifying, 5-10 minutes discussing approach, 20-25 minutes coding, and 5-10 minutes testing. If stuck after 15 minutes of thinking, verbalize your thought process and ask for hints.

What is the importance of time and space complexity analysis?

Complexity analysis is crucial because: 1) Interviewers always ask about it, 2) It helps you choose between approaches, 3) It demonstrates CS fundamentals. Always state both time and space complexity, and be prepared to explain how you derived them.

What is the optimal approach for "Best Time to Buy and Sell Stock with Cooldown"?

For this medium-level Array problem, the key is to identify the core pattern. Analyze the input constraints to determine acceptable time complexity. Consider whether dynamic programming techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

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